We present a novel characterization of the mapping of multiple parallelism forms (e.g. data and model parallelism) onto hierarchical accelerator systems that is hierarchy-aware and greatly reduces the space of software-to-hardware mapping. We experimentally verify the substantial effect of these mappings on all-reduce performance (up to 448x). We offer a novel syntax-guided program synthesis framework that is able to decompose reductions over one or more parallelism axes to sequences of collectives in a hierarchy- and mapping-aware way. For 69% of parallelism placements and user requested reductions, our framework synthesizes programs that outperform the default all-reduce implementation when evaluated on different GPU hierarchies (max 2.04x, average 1.27x). We complement our synthesis tool with a simulator exceeding 90% top-10 accuracy, which therefore reduces the need for massive evaluations of synthesis results to determine a small set of optimal programs and mappings.
